Bayesian chronological modelling in Danish archaeology

Bayesian chronological modelling in Danish archaeologyIn Danish archaeology, radiocarbon dating has become an integrated part of the archaeological toolbox. A certain scepticism towards the accuracy of the method means, however, that it often remains a supplement to archaeological interpretation and other informal dating methods.Bayesian modelling counters this scepticism by combining radiocarbon dating with archaeological observations and other dating methods such as stratigraphy, dendrochronology and numismatic dating. Based on data from sources such as these, a Bayesian model calculates the statistical probability distribution of individual radiocarbon dates. Because the modelled dates consider all available information relating to the samples and their context, they can produce more accurate, robust dates and chronologies than those based on simple calibrated dates. Moreover, through Bayesian modelling, it is possible to estimate the dates of events that cannot be dated directly, such as the beginning or end of a settlement phase.The benefits of implementing Bayesian modelling in Danish archaeology are considerable. However, given that it can seem confusing and difficult to comprehend, in the following we will introduce the method by presenting and discussing some examples of Bayesian modelling.Radiocarbon dates are probabilistic, which means that each radiocarbon measurement holds considerable uncertainty. Each radiocarbon date is expressed as a bell-shaped normal distribution around a median (fig. 1). The date is reported as a radiocarbon age and a standard deviation (e.g. 1690 BP ± 30 years). The uncertainty of the radiocarbon measurement is often increased through calibration to calendar years by matching with the wiggles and plateaus of the calibration curve. Since calibrated radiocarbon dates are distributed around the radiocarbon sample’s actual age, a visual assessment of the calibration plot will often lead to misinterpretation of the date of specific events or the beginning, end, or duration of phases. Bayesian statistics is a way of countering these uncertainties and misinterpretations. In the following, we use the calibration program OxCal. The first example is a fictional case where ten simulated radiocarbon dates, corresponding to known years at 10-year intervals between AD 970 and 1060, are calibrated (fig. 2). From a visual assessment of the calibrated dates, it seems they are contemporaneous since the probability distribution of each individual date is up to 200 years. The wide probability distribution blurs the fact that the events are each ten years apart. If we add the prior information that the events form a sequence in which sample A is older than sample B etc., the modelled dates then display narrower probability distributions (fig. 3). These are called posterior probability distributions. A so-called Boundary is added to the model to limit the sequence and mark the first and last non-dated event, since it is unlikely that a sample representing the first and last event in a sequence has been taken.Stratigraphic information is termed an informative prior, while an uninformative prior represents a situation where the only information about the samples is that they belong to the same phase. Uninformative priors are illustrated by five samples from the postholes of an Iron Age house. The house had been in use for 30 years. The simulated dates are then placed within a 30-year period (fig. 4). Again, the unmodelled dates blur the actual duration of the use-phase of the house. The prior information that the samples are interpreted as being contemporaneous is now added to the model using the Phase command. The model then estimates when the use of the house began and when it ended. In OxCal, the Agreement Index, A, is an indicator of the match between the data and the model. It is based on the overlap of the calibrated probability distribution and the posterior distributions. An Agreement Index below 60% is an indication of a problematic sample. An Agreement Index is also calculated for the whole model (Amodel).In a more complex example, stratigraphical information regarding the Iron Age house is added. Samples from a stratigraphically younger house and a younger pit are added to the model as two phases in a sequence (fig. 5). The three samples from the pit are regarded as being contemporaneous, and the ‘combine’ command is used.In the simulated example, five samples are enough to create a robust model for a house’s use-phase. But the number of samples needed also depends on where on the calibration curve the dates end up, and a small number of samples may be compensated for by strong priors. The following example is not simulated but from the excavation of an Iron Age house. Three samples were taken from a roof-bearing post. The samples were taken from growth rings spaced, respectively, 10 and 12 years apart. The charred remains of hazel wattle were found between the stones in the cobbled floor (fig. 6). The hazel was interpreted as the remains of the wattle walls of the excavated house. Two ditches surrounding the house were sampled, one of which was stratigraphically older than the other. The samples and the prior information were combined in a Bayesian model. The house’s date was narrowed from 158-8 BC to 91-6 BC (fig. 7).The final example is from the excavation of a medieval house in a town (fig. 10). Five samples from the floors in the house’s basement were added in a Bayesian model (fig. 9). The floors superseded each other. Moreover, two dendrochronological samples from latrine barrels, older than the house, and samples from the barrels’ content were added to the model (fig. 10). Based on the archaeological interpretation, the use of the house was dated to between AD 1250 and 1450. However, the model showed that it was more likely to have been in use between AD 1413 and 1487 (fig. 11). This new date suggests that the potsherds found in the floor layers were redeposited. These examples demonstrate the considerable potential of Bayesian modelling. However, they also show that it is essential to exercise great care in constructing the model and providing a thorough account of the archaeological priors. It may be necessary to create several models to test the priors’ robustness or to test different priors. Bayesian modelling presents a systematic and formalised way to test various archaeological interpretations.Bayesian chronological modelling in Danish archaeologyIn Danish archaeology, radiocarbon dating has become an integrated part of the archaeological toolbox. A certain scepticism towards the accuracy of the method means, however, that it often remains a supplement to archaeological interpretation and other informal dating methods.Bayesian modelling counters this scepticism by combining radiocarbon dating with archaeological observations and other dating methods such as stratigraphy, dendrochronology and numismatic dating. Based on data from sources such as these, a Bayesian model calculates the statistical probability distribution of individual radiocarbon dates. Because the modelled dates consider all available information relating to the samples and their context, they can produce more accurate, robust dates and chronologies than those based on simple calibrated dates. Moreover, through Bayesian modelling, it is possible to estimate the dates of events that cannot be dated directly, such as the beginning or end of a settlement phase.The benefits of implementing Bayesian modelling in Danish archaeology are considerable. However, given that it can seem confusing and difficult to comprehend, in the following we will introduce the method by presenting and discussing some examples of Bayesian modelling.Radiocarbon dates are probabilistic, which means that each radiocarbon measurement holds considerable uncertainty. Each radiocarbon date is expressed as a bell-shaped normal distribution around a median (fig. 1). The date is reported as a radiocarbon age and a standard deviation (e.g. 1690 BP ± 30 years). The uncertainty of the radiocarbon measurement is often increased through calibration to calendar years by matching with the wiggles and plateaus of the calibration curve. Since calibrated radiocarbon dates are distributed around the radiocarbon sample’s actual age, a visual assessment of the calibration plot will often lead to misinterpretation of the date of specific events or the beginning, end, or duration of phases. Bayesian statistics is a way of countering these uncertainties and misinterpretations. In the following, we use the calibration program OxCal. The first example is a fictional case where ten simulated radiocarbon dates, corresponding to known years at 10-year intervals between AD 970 and 1060, are calibrated (fig. 2). From a visual assessment of the calibrated dates, it seems they are contemporaneous since the probability distribution of each individual date is up to 200 years. The wide probability distribution blurs the fact that the events are each ten years apart. If we add the prior information that the events form a sequence in which sample A is older than sample B etc., the modelled dates then display narrower probability distributions (fig. 3). These are called posterior probability distributions. A so-called Boundary is added to the model to limit the sequence and mark the first and last non-dated event, since it is unlikely that a sample representing the first and last event in a sequence has been taken.Stratigraphic information is termed an informative prior, while an uninformative prior represents a situation where the only information about the samples is that they belong to the same phase. Uninformative priors are illustrated by five samples from the postholes of an Iron Age house. The house had been in use for 30 years. The simulated dates are then placed within a 30-year period (fig. 4). Again, the unmodelled dates blur the actual duration of the use-phase of the house. The prior information that the samples are interpreted as being contemporaneous is now added to the model using the Phase command. The model then estimates when the use of the house began and when it ended. In OxCal, the Agreement Index, A, is an indicator of the match between the data and the model. It is based on the overlap of the calibrated probability distribution and the posterior distributions. An Agreement Index below 60% is an indication of a problematic sample. An Agreement Index is also calculated for the whole model (Amodel).In a more complex example, stratigraphical information regarding the Iron Age house is added. Samples from a stratigraphically younger house and a younger pit are added to the model as two phases in a sequence (fig. 5). The three samples from the pit are regarded as being contemporaneous, and the ‘combine’ command is used.In the simulated example, five samples are enough to create a robust model for a house’s use-phase. But the number of samples needed also depends on where on the calibration curve the dates end up, and a small number of samples may be compensated for by strong priors. The following example is not simulated but from the excavation of an Iron Age house. Three samples were taken from a roof-bearing post. The samples were taken from growth rings spaced, respectively, 10 and 12 years apart. The charred remains of hazel wattle were found between the stones in the cobbled floor (fig. 6). The hazel was interpreted as the remains of the wattle walls of the excavated house. Two ditches surrounding the house were sampled, one of which was stratigraphically older than the other. The samples and the prior information were combined in a Bayesian model. The house’s date was narrowed from 158-8 BC to 91-6 BC (fig. 7).The final example is from the excavation of a medieval house in a town (fig. 10). Five samples from the floors in the house’s basement were added in a Bayesian model (fig. 9). The floors superseded each other. Moreover, two dendrochronological samples from latrine barrels, older than the house, and samples from the barrels’ content were added to the model (fig. 10). Based on the archaeological interpretation, the use of the house was dated to between AD 1250 and 1450. However, the model showed that it was more likely to have been in use between AD 1413 and 1487 (fig. 11). This new date suggests that the potsherds found in the floor layers were redeposited. These examples demonstrate the considerable potential of Bayesian modelling. However, they also show that it is essential to exercise great care in constructing the model and providing a thorough account of the archaeological priors. It may be necessary to create several models to test the priors’ robustness or to test different priors. Bayesian modelling presents a systematic and formalised way to test various archaeological interpretations